International Journal of Heat and Mass Transfer 82 (2015) 159–169

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

CFD analysis of power-law fluid flow and heat transfer around a confinedsemi-circular cylinder

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.11.0460017-9310/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +36 30 511 3834.E-mail addresses: dhimuamit@rediffmail.com, amitdfch@iitr.ac.in (A. Dhiman),

arambl@uni-miskolc.hu (L. Baranyi).1 Tel.: +91 1332 285890 (O), +91 9410329605 (M).

Anuj Kumar a, Amit Dhiman a,1, László Baranyi b,⇑a Department of Chemical Engineering, Indian Institute of Technology Roorkee, Roorkee 247 667, Indiab Department of Fluid and Heat Engineering, Institute of Energy Engineering and Chemical Machinery, University of Miskolc, 3515 Miskolc-Egyetemváros, Hungary

a r t i c l e i n f o

Article history:Received 11 September 2014Received in revised form 11 November 2014Accepted 11 November 2014

Keywords:Semi-circular cylinderConfined flowPower-law indexMomentum transferHeat transferDrag coefficients and Nusselt number

a b s t r a c t

A numerical analysis using Ansys Fluent was carried out to investigate the forced convection of power-law fluids (power-law index varying from 0.2 to 1.8) around a heated semi-circular cylinder with wallconfinement (or blockage ratio) of 25%, Prandtl number of 50, and Reynolds numbers 1–40. Flow andthermal fields were found to be steady for Re up to 40. The shear-thickening behavior was found to havea higher value of drag coefficient, whereas the shear-thinning behavior had a smaller value of drag coef-ficient when compared with Newtonian fluids in the steady regime. The wake size was found shorter inshear-thickening fluids than Newtonian and shear-thinning fluids. An overall heat transfer rate wascalculated and found to increase with the rise in Reynolds number. The average Nusselt numbers wereobserved higher for shear-thinning fluids than Newtonian and shear-thickening fluids; and the maximumenhancement in the heat transfer was achieved approximately 47% as compared to Newtonian fluids. Thepresent results have also been correlated in terms of wake length, drag coefficient and average Nusseltnumber expressions for various Reynolds numbers and power-law indices studied. In addition, the effectsof blockage ratios ranging from 16.67% to 50% on the engineering output parameters with varying power-law index at Re = 40 were reported.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The viscous flow and heat transfer characteristics of a longsemi-circular bluff body (or a semi-circular cylinder) not only havemany practical applications like electronic cooling, pin type heatexchange systems, thermal processing of foodstuffs, papers, fibroussuspensions and others, but also offer space economy in terms ofthe specific heat transfer area. Also, under suitable settings of flowand heat transfer, most multiphase mixtures (foams, paper pulpsuspensions, emulsions, fiber reinforced resin processing, etc.)and high molecular weight polymeric systems (solutions, blends,melts, etc.) exhibit shear-thinning and or shear-thickening behav-iors [1–3]. Extensive literature suggests that many researchershave investigated numerically and or experimentally the non-Newtonian flow and heat transfer around bluff bodies like circularand square, but only a few corresponding details are available onmomentum and heat transfer over a semi-circular cylinder in spiteof its many engineering applications. The present work is

concerned with the confined flow and heat transfer of power-lawfluids around a semi-circular cylinder at low Reynolds numbers(Re) for the Prandtl number (Pr) of 50. The value of the Prandtlnumber of the order of 50 or so is very frequent in chemical, petro-leum and oil related engineering applications and selected basedon the widespread literature [4–6]. At the outset, it is appropriateto briefly review the relevant studies.

Significant investigations have been reported in the literaturefor Newtonian fluids around an unconfined semi-circular bluffbody. Kiya and Arie [7] investigated numerically the fluid flow pastsemi-circular and semi-elliptical projections attached to a planewall for Re ranging from 0.1 to 100. They reported geometricalshapes of front and rear standing vortices, drag coefficients, pres-sure and shear-stress distributions as functions of Re. Unlike [7],Forbes and Schwartz [8] examined the steady flow over a semi-circular obstacle on the bottom of a stream and the wave resis-tance was calculated at the free surface. In a similar study, Forbes[9] obtained the value of the critical flow by using upstream Froudenumber as a part of the solution in the formulation of Forbes andSchwartz [8] for the flow over a semi-circular obstruction.Experimentally, Boisaubert et al. [10] analyzed the flow over asemi-circular cylinder for flat and round sides facing the flow usinga solid tracer visualization technique for Re = 60–600. They found

Nomenclature

CD total drag coefficient, dimensionlessCDF friction drag coefficient, dimensionlessCDP pressure drag coefficient, dimensionlesscp specific heat of the fluid, J kg�1 K�1

Cp pressure coefficient, dimensionlessCV control volumeD diameter of a semi-circular cylinder, mFD drag force per unit length of the obstacle, N m�1

FDF friction drag force per unit length of the obstacle, N m�1

FDP pressure drag force per unit length of the obstacle,N m�1

h local heat transfer coefficient, W m�2 K�1

H domain height, mI2 second invariant of the rate of the strain tensor, s�2

k thermal conductivity of the fluid, W m�1 K�1

L length of the domain, mLr/D recirculation length, dimensionlessm power-law consistency index, Pa sn

n power-law index, dimensionlessnx x-component of the direction vector normal to the sur-

face of the obstacle, dimensionlessnS direction vector normal to the surface of the obstacle,

dimensionlessNuL local Nusselt number, dimensionless

Nu average Nusselt number, dimensionlessp pressure, PaPr Prandtl number, dimensionlessRe Reynolds number, dimensionlessS arc length along the cross-section of a semi-circular

cylinderT absolute temperature, KT1 temperature of the fluid at the channel inlet, KTw constant wall temperature at the surface of the obstacle,

KUavg average velocity of the fluid at the inlet, m s�1

Vx, Vy x- and y-components of the velocity, m s�1

x, y streamwise and transverse coordinates, mXd downstream distance, mXu upstream distance, m

Greek symbolsb blockage ratio, D/H, dimensionlessd smallest size of the CV clustered around the obstacle, mh dimensionless temperature, (T � T1)/(Tw � T1)g viscosity, Pa sq fluid density, kg m�3

e component of rate of strain tensor, s�1

s extra stress tensor, Pa

160 A. Kumar et al. / International Journal of Heat and Mass Transfer 82 (2015) 159–169

critical Reynolds numbers for the onset of vortex shedding as 140and 190 for flat and curved surfaces, respectively. They also studiedflow behavior by introducing a splitter plate behind the roundedbody configuration and suggested the suitability of this arrange-ment as a flow-controlling device [11,12].

Kotake and Suwa [13] investigated the variation of stagnationpoints and the behavior of vortices in the rear of a semi-circularcylinder in the uniform shear flow by the visualization techniqueof the hydrogen bubble method. They showed that in case of shearflow, there was no vortex on the side with the faster main streamspeed and the vortices were generated only on the slower speedside. Iguchi and Terauchi [14] studied the three kinds of non-circular cylinders (e.g. semi-circular, triangular and rectangular)to detect the shedding frequency of Kármán vortex streets forvelocity lower than 10 cm/s. A triangular cylinder was found tomeet the requirement most adequately as long as minimumdetectable velocity was approximately 5 cm/s in the direction offlow approaching the triangular cylinder. Sophy et al. [15] exam-ined the flow past a semi-circular cylinder with curved surface fac-ing the flow and found the flow to be unsteady at Re = 65. Theyobtained the corresponding Strouhal number as 0.166, whichwas 7% larger than that of a circular cylinder at the correspondingtransition. Coutanceau et al. [16] explained not only the way offormation of the initial wake vortices (primary and secondaryvortices), but also their development with time behind a shortcylindrical semi-circular shell. They reported about regime wherestructure changes occurred beyond the first phase of developmentwhen Re was between 120 and 140.

At high Re = 3500, Koide et al. [17] investigated the synchroni-zation of Kármán vortex shedding by giving a controlled cross-flow oscillation to circular, semi-circular and triangular cylinders.They showed that the synchronization region was almost thesame for the three cylinders in spite of the different behaviorsof separation point. Koide et al. [18] also experimentally investi-gated the influence of the cross-sectional configuration of acylindrical body on Kármán vortex excitation by using the samecylinders. They found that Kármán vortex excitation appears on

all the three cylinders, but the oscillation behavior was drasticallydifferent among them.

Recently, Chandra and Chhabra [19] reported the unconfinedflow and heat transfer over a semi-circular cylinder immersed inNewtonian fluids for Re = 0.01–39.5 and Pr = 0.72–100. The criticalReynolds numbers for the wake formation and for the onset of vor-tex shedding were identified as 0.55–0.6 and 39.5–40, respectively.They showed that the total drag was dominated by the pressurecontribution even at low Re. Similarly, the heat transfer resultsconform to the expected positive dependence on Re and Pr. Bhin-der et al. [20] numerically investigated the forced convective heattransfer characteristics past a semi-circular cylinder at incidencefor Re = 80–180 and Pr = 0.71. They showed that the increase inangle of incidence increases streamline curvature. Strouhal num-ber showed a decreasing trend up to certain values of angle ofattack and thereafter it increases marginally. A correlation ofStrouhal number as a function of Re and angle of attack was estab-lished. In a similar study, Chatterjee et al. [21] simulated the forcedconvection heat transfer from a semi-circular cylinder in an uncon-fined flow regime for Re = 50–150 and Pr = 0.71. They consideredtwo different configurations of the semi-circular cylinder; onewhen the curved surface facing the flow and the other when theflat surface facing the flow. They found significant differences inthe global flow and heat transfer quantities for the two configura-tions studied, and concluded that the heat transfer rate wasenhanced substantially when the curved surface was facing theflow rather than the flat surface. More recently, Sukesan and Dhi-man [22] investigated the cross-buoyancy mixed convectionaround a confined semi-circular cylinder at low Re (1–40) for vary-ing Pr (0.71–50) and blockage ratio (16.67–50%).

As far as the flow of non-Newtonian fluids around a semi-circular cylinder is concerned, Chandra and Chhabra [23]delineated the onset of flow separation from the surface of theunconfined semi-circular cylinder and the onset of the laminarvortex shedding regime for power-law fluids. They showed thatirrespective of the type of fluid behavior (n = 0.2–1.8), both thesetransitions occur at the value of Re lower than that for a circular

A. Kumar et al. / International Journal of Heat and Mass Transfer 82 (2015) 159–169 161

cylinder. Likewise, Chandra and Chhabra [24] simulated the flowand heat transfer of power-law fluids over a semi-circular cylinderin the steady regime for the range of physical control parameters atRe = 0.01–30, Pr = 1–100 and n = 0.2–1.8. They showed that theheat transfer rate in shear-thinning (or pseudo-plastic) fluids(n < 1) can be enhanced by up to 60–70% under appropriate condi-tions, whereas shear-thickening (or dilatant) fluids (n > 1) slowdown the heat transfer rate. Simple expressions for recirculationlength, surface pressure and Nusselt number were also derived.They subsequently examined the effects of mixed [25] and natural[26,27] convection around a semi-circular cylinder to power-lawfluids in the unconfined steady regime. In a recent study, Tiwariand Chhabra [28] investigated the flow of power-law fluids pasta semi-circular cylinder with its flat face oriented upstream forRe = 0.01–25, n = 0.2–1.8 and Pr = 0.72–100. The critical Reynoldsnumber for the onset of wake formation for a semi-circular cylin-der with its curved face oriented in the upstream direction is foundlower than that of a semi-circular cylinder with its flat faceoriented in the upstream direction. In contrast, the criticalReynolds number for the onset of vortex shedding for a semi-circular cylinder with its curved face oriented in the upstreamdirection is found a little higher than that of a semi-circularcylinder with its flat face oriented in the upstream direction.

While the literature has dealt fairly extensively with momen-tum and heat transfer phenomena around a semi-circular cylinderfor Newtonian fluids, studies dealing with non-Newtonian fluidsare much rarer, and to the best of our knowledge, none have trea-ted the case of a semi-circular cylinder in a channel, despite themany engineering applications [1–3]. Accordingly, the presentwork aims to fill this gap in the literature on the confined flowand heat transfer around a semi-circular cylinder at low Re forshear-thinning, Newtonian and shear-thickening behaviors.

2. Problem description

The channel confined flow was approximated by considering thelaminar and incompressible flow (fully developed velocity profile)of power-law fluids across an infinitely long semi-circular cylinder(of diameter D) between the two parallel plane solid (adiabatic)walls, as shown schematically in Fig. 1. The power-law fluid at atemperature T1 at the inlet exchanges heat with the isothermalsemi-circular cylinder whose surface was maintained at a temper-

Fig. 1. Schematic of the confined flow and heat transfer

ature Tw such that Tw > T1. The length and the height of the compu-tational domain were identified as L (=Xu + Xd) and H, respectively inaxial and lateral dimensions. The semi-circular cylinder was locatedin the middle, i.e. at the center-line, at an upstream distance of Xu

from the inlet and at a downstream distance of Xd from the outlet.It is worthwhile to mention that the output parameters such as dragcoefficients and wake length with power-law fluids (n = 0.5–2)show an entirely different nature at high value of blockage (e.g.b = D/H = 25%) at low Re (1–45) for the flow around a cylinder ofsquare cross-section [29]. For instance, total drag coefficient alwaysincreases with increasing power-law index for b = 25% at a fixed Re;however, a mixed trend of increase or decrease with power-lawindex is reported for low blockages (b = 16.67% and 12.5%).Furthermore, Zdravkovich [30] reported for the flow aroundcircular cylinders that for 0.1 < b < 0.6 the blockage modifies theflow and the correction of data is necessary. However, this roughclassification given in [30] is applicable for transition in shearlayers, transition in boundary layers and fully turbulent states offlow. Therefore, the detailed flow and heat transfer investigationshave been carried here at a blockage ratio of 25% based on therelevant studies in the confined domain [29–34].

The temperature difference between the surface of the semi-circular cylinder and the streaming liquid (Tw � T1) was kept low(=2 K) in this study so that the variation of the physical properties,notably density and viscosity with temperature could be neglected.Thus, the thermo-physical properties of the streaming liquid wereassumed to be independent of the temperature. The viscousdissipation effects were also assumed to be negligible. Theabove-mentioned two assumptions restrict the applicability ofthese results to the situations for small temperature difference orwhere the temperature difference was not too significant and formoderate viscosity and or shearing levels.

The compact forms of continuity, momentum and energy equa-tions for the present flow system are given below by Eqs. (1)–(3),respectively [2,3,35]:

r � V ¼ 0; ð1ÞqðV � rV � fÞ � r � r ¼ 0; ð2ÞqcpðV � rTÞ � kr2T ¼ 0: ð3Þ

Here V and f are defined as velocity (Vx and Vy were the componentsin Cartesian coordinates) and body force, respectively. The stress

in a channel with a built-in semi-circular cylinder.

162 A. Kumar et al. / International Journal of Heat and Mass Transfer 82 (2015) 159–169

tensor (r) was defined as the sum of the isotropic pressure (p) andthe extra stress tensor (s), i.e. r = �pI + s.

For incompressible fluids, the extra stress tensor was definedusing viscosity (g) and component of the rate of strain tensore(V) as

s ¼ 2geðVÞ; ð4Þ

where

eðVÞ ¼ 12rVð Þ þ rVð ÞT

h iand g ¼ m

I2

2

� �ðn�1Þ=2

; where

I2 ¼ 2 e2xx þ e2

yy þ e2xy þ e2

yx

� �:

The physical boundary conditions for the flow system underconsideration can be written as follows:

� At inlet: a fully developed flow was assumed,

Vx ¼2nþ 1nþ 1

� �Uavg 1� 1� 2y

H

��������

� �ðnþ1Þ=n" #

for 0 � y � H;

Vy ¼ 0 and T ¼ T1: ð5Þ

� On top/bottom adiabatic channel wall: no-slip condition wasapplied,

Vx ¼ 0; Vy ¼ 0 and@T@y¼ 0: ð6Þ

� On the semi-circular cylinder: no-slip condition was used,

Vx ¼ 0; Vy ¼ 0 and T ¼ Tw: ð7Þ

� At outlet: it is located sufficiently far downstream from thesemi-circular cylinder with a zero diffusion flux,

@Vx

@x¼ 0;

@Vy

@x¼ 0 and

@T@x¼ 0: ð8Þ

The mathematical expressions of various engineering parame-ters utilized in this study are given as follows:

� Reynolds and Prandtl numbers for non-Newtonian power-lawfluids were defined here as:

Re ¼qDnU2�n

avg

mand Pr ¼ mcp

kUavg

D

� �n�1

:

� Recirculation (or wake) length (Lr/D) was measured from the rearstagnation point to the point of reattachment for the near closedstreamline on the line of symmetry in the downstream region.� Overall drag coefficient (CD) and its components (CDP and CDF)

were mathematically defined as:

CD ¼FD

12 qU2

avgD¼ CDP þ CDF ;

where

CDP ¼FDP

12 qU2

avgD¼Z

SCpnxdS and CDF ¼

FDF

12 qU2

avgD

¼ 2Re

ZS

s � nSð ÞdS:

� The local Nusselt number on the surfaces (curved and flat one)of the semi-circular cylinder was evaluated for the constantwall temperature as NuL = hD/k = �oh/onS. These local valueson each surface were then averaged to obtain the averaged Nus-selt number of the semi-circular cylinder.

Nu ¼ 1S

ZS

NuLdS:

Numerical details

3.

The numerical computations were carried out by solving thegoverning Eqs. (1)–(3) along with the boundary conditions (5)–(8) for the primitive variables, i.e. velocity (Vx and Vy), pressure(p) and temperature (T) fields by using a commercial computa-tional fluid dynamics solver Ansys Fluent [36]. The two-dimen-sional, steady/unsteady, laminar, segregated solver was employedto solve the incompressible flow on the non-uniform collocatedgrid. The second-order upwind scheme was used to discretize con-vective terms, while the diffusive and the non-Newtonian termswere discretized by the central difference scheme. The SIMPLEscheme was used for solving pressure–velocity decoupling. Theconstant density and the non-Newtonian power-law viscositymodel were used. The fully developed velocity profile at the chan-nel inlet (see Eq. (5)) was incorporated by using the user-definedfunctions obtainable in Ansys Fluent [36]. Ansys Fluent solvedthe system of algebraic equations by using the Gauss–Siedelpoint-by-point iterative method in conjunction with the algebraicmulti-grid method solver. The absolute convergence criteria of10�10 for the continuity and x- and y-components of the velocityand of 10�15 for the energy were prescribed in the steady regime.In spite of this, the absolute convergence criteria of 10�15 each forthe continuity, x- and y-components of the velocity and energywere used in the unsteady regime.

The computational grid was generated by using Gambit and itszoomed view around the semi-circular obstacle is shown in Fig. 2for b = 25%. A very fine grid of cell size (with number of control vol-umes (CVs) on the semi-circular cylinder equal to 340) of 0.01Dwas clustered around the semi-circular obstacle and near the chan-nel walls; however, the largest grid size of 0.5D is utilized awayfrom the semi-circular cylinder and channel walls.

The effect of the grid size on the dimensionless output param-eters such as drag coefficient (CD) and average Nusselt number(Nu) was investigated at three grid structures (symbolically rep-resented as G1, G2 and G3 with 250, 340 and 400 CVs prescribedon the surfaces of a semi-circular cylinder) for Re = 40, Pr = 50,b = 25% and n = 0.2, 1, 1.8 (Table 1). The relative percent differ-ences in the values of CD and Nu for the grid G1 (250 CVs) withrespect to the values at the grid G3 (400 CVs) for n = 0.2 werefound to be about 3% and 2%, respectively. For n = 1, the relativepercent differences in the values of CD and Nu for the grid G1(250 CVs) with respect to the values at the grid G3 (400 CVs)were found to be about 2.2% and 2.5%, respectively. Similarly,at n = 1.8, the relative differences in the values of CD and Nufor the grid G1 (250 CVs) with respect to the values at the gridG3 (400 CVs) were found to be about 1.3% and 3%, respectively.On the other hand, the corresponding differences in the valuesof CD and Nu for the grid G2 (340 CVs) with respect to the valuesat the finest grid G3 (400 CVs) were found to be only 1% and lessthan 1.1%, respectively. Thus, the grid G2 was used to generatefurther results.

The domain dependence study was conducted to determine theeffects of upstream (Table 2) and downstream (Table 3) distanceson the dimensionless output parameters as follows. The influenceof the upstream distance for b = 16.67% on the values of physicalparameters was investigated for Xu = 30D, 45D and 60D forRe = 40 and Pr = 50 at n = 0.2, 1 and 1.8 (Table 2). For n = 0.2, 1and 1.8, the relative percent differences in the values of CD andNu for Xu/D = 45 with respect to Xu/D = 60 were found to be lessthan 1% for both CD and Nu. Similarly, the influence of thedownstream distance on the values of physical parameters wasinvestigated for Xd = 100D, 120D and 140D for the same values ofRe, Pr and n (Table 3). The relative percent differences were foundto be less than 0.6% for both CD and Nu for Xd/D = 120 with respectto Xd/D = 140. For b = 25%, the relative percent differences in the

x

y

40 42 44 46 48 500

1

2

3

4

Fig. 2. Magnified view of the grid around a semi-circular cylinder for b = 25%.

Table 1Grid dependence study for Re = 40, n = 0.2, 1, 1.8 and Pr = 50.

Grid details CD Nu

d/D CVs on semi-circular cylinder Total number of cells in domain n

0.2 1 1.8 0.2 1 1.8

G1 0.1 250 43436 1.961 4.088 6.457 25.997 18.880 15.552G2 0.01 340 103579 1.905 4.001 6.650 26.588 19.154 15.852G3 0.008 400 127691 1.905 4.000 6.649 26.597 19.379 16.028

Table 2Effect of upstream distance on the dimensionless output parameters for Re = 40 andPr = 50 at different values of b and n.

Xu/D CD Nu

n

0.2 1 1.8 0.2 1 1.8

b = 16.67%30 1.655 3.336 5.047 24.796 18.482 15.27645 1.653 3.336 5.047 24.776 18.491 15.27460 1.637 3.335 5.047 24.647 18.512 15.271

b = 25%30 1.887 4.002 6.630 26.605 19.160 15.85245 1.905 4.001 6.650 26.588 19.154 15.85160 1.915 4.002 6.650 26.361 19.138 15.847

b = 50%30 3.762 9.329 36.611 31.220 22.215 16.44545 3.747 9.321 36.572 31.469 22.283 16.46760 3.726 9.319 36.110 31.554 22.418 16.474

Table 3Effect of downstream distance on the dimensionless output parameters for Re = 40and Pr = 50 at different values of b and n.

Xd/D CD Nu

n

0.2 1 1.8 0.2 1 1.8

b = 16.67%100 1.652 3.336 5.047 24.710 18.410 15.289120 1.653 3.336 5.047 24.776 18.491 15.274140 1.656 3.332 5.047 24.901 18.502 15.291

b = 25%100 1.879 4.002 6.647 26.606 19.140 15.868120 1.905 4.002 6.650 26.588 19.154 15.851140 1.914 4.001 6.645 26.572 19.169 15.848

b = 50%100 3.727 9.319 36.421 31.579 22.310 16.593120 3.747 9.321 36.572 31.469 22.283 16.467140 3.751 9.319 36.624 31.239 22.110 16.450

A. Kumar et al. / International Journal of Heat and Mass Transfer 82 (2015) 159–169 163

values of CD and Nu for Xu/D = 45 were found to be less than 0.9%with respect to the values at Xu/D = 60 for n = 0.2, 1, and 1.8(Table 2). Whereas, the corresponding differences in the down-stream distances for Xd/D = 120 were found to be less than 0.5%for both the output parameters with respect to Xd/D = 140 (Table 3).Along the same line, for b = 50%, the relative differences in the val-ues of both CD and Nu for Xu/D = 45 were found to be less than 1.3%of the corresponding values at Xu/D = 60, whereas the correspond-ing differences in the downstream distances for Xd/D = 120 werefound to be less than 0.8% with respect to Xd/D = 140 for both theoutput parameters (Table 3). Thus, the dimensionless upstreamand downstream distances of 45 and 120, respectively, were foundadequate for the present results to be free from end effects.

4. Results and discussion

The confined flow of power-law fluids (0.2 < n < 1.8) was simu-lated here for Re = 1–40 and for the fixed blockage ratio of 25%

[31–34] at the Prandtl number of 50 [4–6]. Additionally, the effectsof blockage ratios of 16.67%, 25%, and 50% on the engineering out-put parameters with varying power-law index at the maximumvalue of Reynolds number investigated here (Re = 40) werereported. It is to be noted that Chandra and Chhabra [19] foundthe transition from a steady to a time-periodic regime betweenRe = 39.5 and 40 for the flow around a semi-circular cylinder inthe unconfined domain. Therefore, time-dependent computationshave also been carried out for the present confined flow configura-tion with a built-in semi-circular cylinder at Re = 40 for theextreme values of blockage ratios (16.67% and 50%) and power-law indices (0.2 and 1.8) studied. The flow and the thermal fieldswere found to be steady for the entire range of settings coveredin this study. This is attributed to the fact that as the blockageincreases the flow tends to stabilize, because the walls are nearerto the semi-circular cylinder.

Unfortunately no literature is available for the confined flowof power-law fluids across a semi-circular cylinder, so the

164 A. Kumar et al. / International Journal of Heat and Mass Transfer 82 (2015) 159–169

benchmarking of the present numerical solution procedure wasmade for the unconfined case in the steady regime for n = 0.2–1.8and Re = 1–30 by using the identical computational domain ofChandra and Chhabra [25]. Excellent agreement can be seenbetween the present results and the values reported in [25](Table 4). For instance, the maximum differences in the values oftotal drag coefficients and average Nusselt numbers were foundto be less than 1.3% and about 1.6%, respectively. Additional bench-marking of the numerical methodology employed can be found inour recent study for the confined power-law flow and heat transferaround an inclined square bluff body [6].

4.1. Flow and thermal patterns

Flow structures in the vicinity of the long semi-circular obstaclewere shown by streamline contours at the blockage ratio of 25% forthe extreme and the intermediate values of Re (1, 20 and 40) and n(0.2, 1 and 1.8) in Fig. 3a–i. The streamline contours showed a con-dition of no flow separation from the surface of the semi-circularcylinder at Re = 1 for the entire range of n studied, due to the dom-inant viscous force at this low value of Re. The flow separationoccurred on increasing the value of Re (>1) from the rear or the flatsurface of the semi-circular cylinder. Two standing symmetric vor-tices, rotating in opposite directions, were observed behind thesemi-circular obstacle in the steady regime, as can be seen inFig. 3. As expected, the size of these vortices increases with anincrease in Re for the constant value of n. But the size of thesevortices was found to decrease as the fluid behavior changed fromshear-thinning to shear-thickening for the constant Re. These find-ings were consistent with the power-law studies reported on theconfined circular [31,37] and square [5,29] cylinders in the steadyregime.

Thermal patterns provide knowledge of the detailed tempera-ture field around an obstacle, which can be significant in thedesigning of a variety of heat exchange systems particularly inthe processing of temperature sensitive materials such as food-stuffs, personal care products and others. Fig. 4a–i display the rep-resentative isotherm contours around the semi-circular cylinder inthe steady regime at Pr = 50. As expected, the curved (or upstream)surface of the semi-circular cylinder had the maximum clusteringof isotherms and the minimum on the flat (or downstream) sur-face. As a result, higher rate of heat transfer was observed fromthe curved surface than the flat one. The increased value of Refor the constant value of n resulted in the increase in the temper-ature gradient around the obstacle due to the thinning of momen-tum and thermal boundary layers and increase in circulation of

Table 4Comparison of present drag and Nusselt number results for the flow around a semi-circula

Re n Present work

CD

1 0.2 26.2161 10.0501.8 5.730

5 0.2 5.5851 3.8821.8 3.149

10 0.2 3.0781 2.7101.8 2.529

30 0.2 1.4281 1.6901.8 1.805

large amount of fluid, thus an increase in the heat transfer ratewas observed. These findings were consistent with the power-law studies reported on the confined heated circular [38] andsquare [5,39] cylinders in a channel at low Re. On the other hand,decomposing of steady temperature fields was observed with theincrease in shear-thickening behavior for the fixed value of Re;however, these effects were more prominent at the high value ofRe. Overall, the heat transfer rate was enhanced by the decreasein n for the fixed Re. The trends observed here were stronglydependent on the value of Re.

4.2. Wake length

The phenomenon of flow recirculation in the rear of the obsta-cle was often visualized in terms of the recirculation (or wake)length. Fig. 5 displays the dependence of the recirculation length(Lr/D) on Re and n at the blockage ratio of 25%. No flow separationwas observed at Re = 1 (i.e. the wake length is zero here) for anyvalue of n studied (Figs. 3 and 5), but at Re = 5 the separation ofboundary layer was observed for all values of n. As expected, thewake region increased with the increase in Re for the fixed valueof n. The length of the recirculation zone increased when movingfrom shear-thickening to shear-thinning nature (or when decreas-ing the value of n) for the fixed value of Re. These findings wereconsistent with the power-law studies available on the steady flowaround confined circular [37] and square [5,29] cylinders.

The present wake length results further can be approximatedsomewhat better by the following expression for the settings1 < Re 6 40 and 0.2 6 n 6 1.8 at b = 25%,

Lr=D ¼ Re½1:2þ0:0315ðlnðnÞÞ2 � exp½ð�2:8405� 0:1212n3Þ� ð0:0142n�0:0502ÞRe�: ð9Þ

Eq. (9) has the average deviation of about 5.5% and the maximumdeviation of less than 10%, when compared with the present com-puted results.

4.3. Overall drag coefficient and its components

Fig. 6a shows the dependence of the pressure drag coefficient(CDP) for a semi-circular cylinder on Re and n at b = 25%. For a fixedvalue of Re, the value of the pressure drag coefficient decreased asthe fluid behavior changes from shear-thickening to shear-thinning. The influence of n was found more prominent at lowRe and its effect on the pressure drag gradually diminished as Rewas increased, a similar trend was seen for spheroid and spherical

r cylinder with literature [25] at various values of Re and n in the unconfined domain.

Chandra and Chhabra [25]

Nu CD Nu

4.613 26.020 4.6462.798 10.170 2.8362.675 5.782 2.665

8.829 5.611 8.9755.397 3.854 5.4714.897 3.189 4.952

12.645 3.104 12.7717.465 2.721 7.5776.829 2.541 6.893

23.069 1.432 23.06513.581 1.696 13.75211.948 1.817 11.882

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Fig. 3. Streamlines for Re = 1, 20 and 40 for n = 0.2, 1 and 1.8 in the steady regime for b = 25%.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Fig. 4. Isotherm contours for Re = 1, 20 and 40 for n = 0.2, 1 and 1.8 in the steady regime for b = 25%.

A. Kumar et al. / International Journal of Heat and Mass Transfer 82 (2015) 159–169 165

Fig. 5. Variation of wake length (Lr/D) with Re at different values of n in the steadyregime for b = 25%.

166 A. Kumar et al. / International Journal of Heat and Mass Transfer 82 (2015) 159–169

particles [1,2] where inertial forces overshadowed the viscousforces with increasing Re, hence independent of the type of fluid(shear-thinning, shear-thickening and Newtonian). Similarly, at afixed value of Re, the dependence of the friction drag coefficient(CDF) was increased with increasing n as the viscous force increaseswith increasing n (Fig. 6b). When Re was gradually increased, thedependence of the friction drag coefficient on n decreases. Broadly,the larger the Re the weaker the dependence on n. These trends

(a)

(c)

Fig. 6. (a–c) Variation of individual and total drag coefficients (CDP, CDF, CD) an

were clearly due to the non-linear nature of the viscosity overthe surface of the semi-circular cylinder [24]. The viscosity forthe shear-thickening fluids becomes very large as the shear ratedecreases and hence it tends to infinity far-away from the semi-circular cylinder. Therefore, the viscous effects dominate far-awayfrom the semi-circular cylinder. Along the same line, the depen-dence of the total drag coefficient(CD) on n was seen to be strongerat low values of Re, thus a significant variation in the drag valuescould be seen in shear-thickening, Newtonian and shear-thinningfluids (Fig. 6c). The value of CD increased when n increases from0.2 to 1.8. These trends were qualitatively consistent with theliterature values for the unconfined power-law flow past asemi-circular cylinder in the steady regime [24]. The total dragcoefficient over the entire range of Re and n studied can berepresented by the following correlation:

CD ¼ 0:7445þ 2:1n� 0:1503n3 þ ð4:7196þ 18:3703

� expðnÞÞ=Reþ ð3:0881� 15:8919n2 þ 3:84n3Þ

� lnðReÞ=Re2: ð10Þ

Eq. (10) has an average deviation of less than 1% and the maximumdeviation of about 3.2% with the present computed results.

Further examination of the present results in terms of the var-iation of the drag ratio (CDP/CDF) on Re and n at b = 25% reveals thatthis ratio decreases with the increase in n for the range of param-eters studied in this work (Fig. 6d). Qualitatively, this effect can beexplained as follows: for a fixed value of Re, the viscous forces scaleas Un

avg . For a shear-thickening fluid, as the value of n is graduallyincreased, the viscous forces increase. Similarly, for a shear-thinning fluid, viscous forces increased more steeply. In either case,the CDP/CDF ratio decreased with n, as shown in Fig. 6d. A rapid

(b)

(d)

d (d) drag ratio (CDP/CDF) with Re and n in the steady regime for b = 25%.

Fig. 7. Variation of the average Nusselt number as a function of Re and n at Pr = 50in the steady regime for b = 25%.

A. Kumar et al. / International Journal of Heat and Mass Transfer 82 (2015) 159–169 167

decrease in the drag ratio with an increase in n was seen at low Rein shear-thickening fluids, i.e. strong dependence on n.

4.4. Average Nusselt number

The variation of the average Nusselt number (Nu) for the vary-ing values of Re and n for the value of Prandtl number at b = 25% isdepicted in Fig. 7. It can be seen that the heat transfer rateincreased with the increase in the value of Re for the constant

(a)

(c)

Fig. 8. (a–c) Variation of individual and total drag coefficients (CDP, CDF, CD) and (d) aveRe = 40 and Pr = 50.

value of n. However, as the value of n was increased, for theconstant value of Re, the average Nusselt number decreases. Theaverage Nusselt number is found higher for shear-thinning fluidsthan Newtonian and shear-thickening fluids. The maximumenhancement in the heat transfer is found approximately 47% forshear-thinning fluid (for Re = 30 and n = 0.2) with respect toNewtonian behavior. It is also found that the value of the heattransfer in the confined semi-circular cylinder is always greaterthan that of the unconfined semi-circular cylinder for power-lawfluids in the steady regime [24].

Furthermore, the following simple expressions have beendeveloped to represent the average cylinder Nusselt number asthe function of Re and n. After a thorough study, it was observedthat two different average Nusselt number expressions, namelyEq. (11) for 0.2 6 n 6 1 and Eq. (12) for 1 < n 6 1.8, can be usedsuch that their maximum deviations with the computed resultsfor the entire range of Re covered at b = 25% were less than 5%:

Nu ¼ ð3:3704n=ð0:01þ nÞÞ þ ð1:581=ð1þ 1:232nÞÞRe� ð0:13� 0:3744nÞRe1:5; ð11Þ

Nu ¼ 3:9337� 0:03095nþ ð�0:1737nþ 0:8224ÞRe0:5 lnðReÞ: ð12Þ

Expression (11) has the maximum deviation of less than 3.6%, withthe present computed results and the corresponding average devi-ation is of less than 1.2%. Similarly, Eq. (12) has the maximumand the average deviations of less than 3.9% and less than 1.3%,respectively, with the present computed results.

4.5. Effect of blockage ratio

To present the influence of blockage ratio on the engineeringoutput parameters, Fig. 8a–d depict the variation of individual

(b)

(d)

rage cylinder Nusselt number (Nu) for n = 0.2–1.8 and b = 16.67%, 25%, and 50% at

168 A. Kumar et al. / International Journal of Heat and Mass Transfer 82 (2015) 159–169

and overall drag coefficients and average cylinder Nusselt numberas a function of power-law index for b = 16.67–50% at Re = 40 andPr = 50.

The drag coefficients were found to increase with increasingblockage ratio for the fixed value of the power-law index. Similarly,the drag coefficients increased with increasing power-law indexfor the fixed blockage ratio. The values of drag coefficients werefound most affected between blockages of 25% and 50%, but a smallcorresponding change in the values of drag coefficients wasobserved between the blockages of 25% and 16.67%. For instance,a quite a big change of approximately 550% (at n = 1.8) wasobserved in the values of overall drag coefficients for b = 50% withrespect to the corresponding values at b = 25%. However, a smallchange in the values of overall drag coefficients of approximately32% (at n = 1.8) was observed for b = 25% with respect to thecorresponding values at b = 16.67%. The drag ratio (as defined inSection 4.3) was observed to decrease sharply with increasingpower-law index; however, it increased with increasing blockageratio. In other words, the contribution of the pressure drag coeffi-cient to the overall drag coefficient was always found to be higherthan that of the friction drag coefficient for the range of settingsstudied.

Similarly to the drag coefficients, the rate of heat transfer wasfound to increase with increasing wall confinement for the con-stant value of n because of the increase in the thermal gradients.However, as opposed to the drag coefficients, the average cylinderNusselt number was found to increase as the value of power-lawindex decreased from 1.8 to 0.2 (or as the shear-thinning tendencyincreased) for the constant blockage ratio. The maximum enhance-ment in heat transfer for the highest blockage ratio of 50% withrespect to the lowest blockage ratio of 16.67% used was found tobe approximately 27% at n = 0.2.

5. Conclusions

Effects of power-law fluids in a channel with a built-in longheated semi-circular cylinder were investigated for the various val-ues of Re (1–40) and n (0.2–1.8) at the values of blockage ratio of25% and Prandtl number of 50. Extensive numerical computationsfree from domain and mesh size effects were performed to calcu-late wake length, drag coefficients and average Nusselt number.For a fixed value of Re, the length of the recirculation zonedecreased with an increase in the value of n. Drag coefficients werefound to increase with the increase in the value of n for the fixedRe. Also, the values of individual and total drag coefficients showedthe usual inverse dependence on Re for the fixed value of n. As iswell known, the heat transfer rate increased with the increase inRe for the constant value of n. Similarly, when the value of nincreased the average Nusselt number decreased. The maximumenhancement in heat transfer for shear-thinning fluids was foundapproximately 47% with respect to Newtonian behavior atb = 25%. The correlations of wake length, overall drag coefficientand average Nusselt number were also established. Finally, theeffect of blockage ratio (16.67–50%) on the flow and heat transferoutput parameters was investigated for the varying power-lawindices (0.2–1.8) at the fixed values of Re = 40 and Pr = 50. The dragcoefficients and the average cylinder Nusselt number increasedwith increasing blockage ratio for any value of n. The drag coeffi-cients were found to increase with increasing n, whereas the aver-age cylinder Nusselt number increased as the value of n decreased.Flow and thermal fields were found to be steady for the entirerange of parameters investigated. A more detailed investigationof the effects of blockage ratios on the flow and heat transfercharacteristics at different Prandtl numbers would be the scopefor future research.

Conflict of interest

None declared.

Acknowledgments

The third author gratefully acknowledges the support of theTÁMOP-4.2.1.B-10/2/KONV-2010-0001 project in the frameworkof the New Hungarian Development Plan. The realization of thisproject is supported by the European Union, co-financed by theEuropean Social Fund.

The authors also would like to thank the two anonymousreviewers for their valuable and helpful comments on this work.

References

[1] R.P. Chhabra, Hydrodynamics of non-spherical particles in non-Newtonianfluids, in: N.P. Cheremisinoff, P.N. Cheremisinoff (Eds.), Handbook of AppliedPolymer Processing Technology, Marcel Dekker, New York, 1996. Chapter 1.

[2] R.P. Chhabra, Bubbles, Drops and Particles in Non-Newtonian Fluids, seconded., CRC Press, Boca Raton, FL, 2006.

[3] R.P. Chhabra, Fluid flow and heat transfer from circular and non-circularcylinders submerged in non-Newtonian liquids, Adv. Heat Transfer 43 (2011)289–417.

[4] M. Bouaziz, S. Kessentini, S. Turki, Numerical prediction of flow and heattransfer of power-law fluids in a plane channel with a built-in heated squarecylinder, Int. J. Heat Mass Transfer 53 (2010) 5420–5429.

[5] N. Sharma, A. Dhiman, S. Kumar, Non-Newtonian power-law fluid flow arounda heated square bluff body in a vertical channel under aiding buoyancy,Numer. Heat Transfer A 64 (2013) 777–799.

[6] A. Kumar, A.K. Dhiman, R.P. Bharti, Power-law flow and heat transfer over aninclined square bluff body: effect of blockage ratio, Heat Transfer-Asian Res. 43(2014) 167–196.

[7] M. Kiya, M. Arie, Viscous shear flow past small bluff bodies attached to a planewall, J. Fluid Mech. 69 (1975) 803–823.

[8] L.K. Forbes, L.W. Schwartz, Free-surface flow over a semi-circular obstruction,J. Fluid Mech. 114 (1982) 299–314.

[9] L.K. Forbes, Critical free surface flow over a semi-circular obstruction, J. Eng.Math. 22 (1988) 3–13.

[10] N. Boisaubert, M. Coutanceau, P. Ehrmann, Comparative early development ofwake vortices behind a short semicircular-section cylinder in two oppositearrangements, J. Fluid Mech. 327 (1996) 73–99.

[11] N. Boisaubert, M. Coutanceau, A. Texier, Manipulation of the startingsemicircular cylinder near-wake by means of a splitter plate, J. Flow Visual.Image Process. 4 (1997) 211–221.

[12] N. Boisaubert, A. Texier, Effect of splitter plate on the near-wake developmentof a semicircular cylinder, Exp. Therm. Fluid Sci. 16 (1998) 100–111.

[13] M. Kotake, S. Suwa, Flow visualization around a semi circular cylinder in auniform shear flow, J. Visual. Soc. Jpn. 21 (2001) 95–98.

[14] M. Iguchi, Y. Terauchi, Karman vortex probe for the detection of molten metalsurface flow in low velocity range, ISIJ Int. 42 (2002) 939–943.

[15] T. Sophy, H. Sada, D. Bouard, Calcul de I’ecoulementautour d’un cylindre semi-circulaire par une method de collocation, C.R. Mecanique 330 (2002) 193–198.

[16] M. Coutanceau, C. Migeon, P. Ehrmann, Particulars of the cross and spanwisenear-wake development of a short semicircular-section shell, through thetransition Re-range (60 < Re < 600), J. Visual. 3 (2000) 9–26.

[17] M. Koide, S. Tomida, T. Takahashi, L. Baranyi, M. Shirakashi, Influence of cross-sectional configuration on the synchronization of Karman vortex sheddingwith the cylinder oscillation, JSME Int J., Ser. B 45 (2002) 249–258.

[18] M. Koide, T. Takahashi, M. Shirakashi, Influence of cross-sectionalconfiguration on Karman vortex excitation, J. Comp. Appl. Mech. 5 (2004)297–310.

[19] A. Chandra, R.P. Chhabra, Flow over and forced convection heat transfer inNewtonian fluids from a semi-circular cylinder, Int. J. Heat Mass Transfer 54(2011) 225–241.

[20] A.P.S. Bhinder, S. Sarkar, A. Dalal, Flow over and forced convection heattransfer around a semi-circular cylinder at incidence, Int. J. Heat Mass Transfer55 (2012) 5171–5184.

[21] D. Chatterjee, B. Mondal, P. Halder, Unsteady forced convection heat transferover a semi-circular cylinder at low Reynolds numbers, Numer. Heat TransferA 63 (2013) 411–429.

[22] M.K. Sukesan, A.K. Dhiman, Laminar mixed convection in a channel with abuilt-in semi-circular cylinder under the effect of cross-buoyancy, Int.Commun. Heat Mass Transfer 58 (2014) 25–32.

[23] A. Chandra, R.P. Chhabra, Influence of power-law index on transitionalReynolds numbers for flow over a semi-circular cylinder, Appl. Math. Model.35 (2011) 5766–5785.

[24] A. Chandra, R.P. Chhabra, Momentum and heat transfer characteristics of asemi-circular cylinder immersed in power-law fluids in the steady flowregime, Int. J. Heat Mass Transfer 54 (2011) 2734–2750.

A. Kumar et al. / International Journal of Heat and Mass Transfer 82 (2015) 159–169 169

[25] A. Chandra, R.P. Chhabra, Mixed convection from a heated semi-circularcylinder to power-law fluids in the steady flow regime, Int. J. Heat MassTransfer 55 (2012) 214–234.

[26] A. Chandra, R.P. Chhabra, Laminar free convection from a horizontal semi-circular cylinder to power-law fluids, Int. J. Heat Mass Transfer 55 (2012)2934–2944.

[27] A.K. Tiwari, R.P. Chhabra, Laminar natural convection in power-law liquidsfrom a heated semi-circular cylinder with its flat side oriented downward, Int.J. Heat Mass Transfer 58 (2013) 553–567.

[28] A.K. Tiwari, R.P. Chhabra, Momentum and heat transfer characteristics for theflow of power-law fluids over a semicircular cylinder, Numer. Heat Transfer A66 (2014) 1365–1388.

[29] A.K. Dhiman, R.P. Chhabra, V. Eswaran, Steady flow across a confined squarecylinder: effects of power-law index and of blockage ratio, J. Non-NewtonianFluid Mech. 148 (2008) 141–150.

[30] M.M. Zdravkovich, Flow Around Circular Cylinders: Applications, vol. 2, OxfordUniversity Press, New York, 2003.

[31] S. Bijjam, A.K. Dhiman, CFD analysis of two-dimensional non-Newtonianpower-law flow across a circular cylinder confined in a channel, Chem. Eng.Commun. 199 (2012) 767–785.

[32] H. Abbassi, S. Turki, S.B. Nasrallah, Numerical investigation of forcedconvection in a plane channel with a built-in triangular prism, Int. J. Therm.Sci. 40 (2001) 649–658.

[33] S. Srikanth, A.K. Dhiman, S. Bijjam, Confined flow and heat transfer across atriangular cylinder in a channel, Int. J. Therm. Sci. 49 (2010) 2191–2200.

[34] R. Agarwal, A. Dhiman, Flow and heat transfer phenomena across two confinedtandem heated triangular bluff bodies, Numer. Heat Transfer A 66 (2014)1020–1047.

[35] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, second ed.,Wiley, New York, 2002.

[36] ANSYS User Manual, Ansys, Inc., Canonsburg, PA, 2009.[37] R.P. Bharti, R.P. Chhabra, V. Eswaran, Two-dimensional steady Poiseuille flow

of power-law fluids across a circular cylinder in a plane confined channel: walleffects and drag coefficients, Ind. Eng. Chem. Res. 46 (2007) 3820–3840.

[38] R.P. Bharti, R.P. Chhabra, V. Eswaran, Effect of blockage on heat transfer from acylinder to power-law liquids, Chem. Eng. Sci. 62 (2007) 4729–4741.

[39] A.K. Dhiman, Heat transfer to power-law dilatant fluids in a channel with abuilt-in square cylinder, Int. J. Therm. Sci. 48 (2009) 1552–1563.